Problem: Simplify and expand the following expression: $ \dfrac{5}{2a - 10}- \dfrac{3}{a - 9}- \dfrac{2}{a^2 - 14a + 45} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{5}{2a - 10} = \dfrac{5}{2(a - 5)}$ We can factor the quadratic in the third term: $ \dfrac{2}{a^2 - 14a + 45} = \dfrac{2}{(a - 5)(a - 9)}$ Now we have: $ \dfrac{5}{2(a - 5)}- \dfrac{3}{a - 9}- \dfrac{2}{(a - 5)(a - 9)} $ The least common multiple of the denominators is: $ 2(a - 5)(a - 9)$ In order to get the first term over $2(a - 5)(a - 9)$ , multiply by $\dfrac{a - 9}{a - 9}$ $ \dfrac{5}{2(a - 5)} \times \dfrac{a - 9}{a - 9} = \dfrac{5(a - 9)}{2(a - 5)(a - 9)} $ In order to get the second term over $2(a - 5)(a - 9)$ , multiply by $\dfrac{2(a - 5)}{2(a - 5)}$ $ \dfrac{3}{a - 9} \times \dfrac{2(a - 5)}{2(a - 5)} = \dfrac{6(a - 5)}{2(a - 5)(a - 9)} $ In order to get the third term over $2(a - 5)(a - 9)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{2}{(a - 5)(a - 9)} \times \dfrac{2}{2} = \dfrac{4}{2(a - 5)(a - 9)} $ Now we have: $ \dfrac{5(a - 9)}{2(a - 5)(a - 9)} - \dfrac{6(a - 5)}{2(a - 5)(a - 9)} - \dfrac{4}{2(a - 5)(a - 9)} $ $ = \dfrac{ 5(a - 9) - 6(a - 5) - 4} {2(a - 5)(a - 9)} $ Expand: $ = \dfrac{5a - 45 - 6a + 30 - 4}{2a^2 - 28a + 90} $ $ = \dfrac{-a - 19}{2a^2 - 28a + 90}$